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Linear Algebra

Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling

Chapter 5

Span and Bases - all with Video Answers

Educators

Chapter Questions

01:48

Problem 1

Show that the vectors $v_{1}=(1,1,1), v_{2}=(1,2,3)$, and $v_{3}=(2,-1,1)$ are linearly independent in $\mathbb{R}^{3}$. Write $v=(1,-2,5)$ as a linear combination of $v_{1}, v_{2}$, and $v_{3}$.

Manisha Sarker
Manisha Sarker
Numerade Educator
02:38

Problem 2

Consider the complex vector space $V=\mathbb{C}^{3}$ and the list $\left(v_{1}, v_{2}, v_{3}\right)$ of vectors in $V$, where
$$v_{1}=(i, 0,0), v_{2}=(i, 1,0), v_{3}=(i, i,-1)$$
(a) Prove that $\operatorname{span}\left(v_{1}, v-2, v_{3}\right)=V$.
(b) Prove or disprove: $\left(v_{1}, v_{2}, v_{3}\right)$ is a basis for $V$.

Cory Glover
Cory Glover
Numerade Educator
01:18

Problem 3

Determine the dimension of each of the following subspaces of $\mathbb{F}^{4}$.
(a) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=0\right\}$.
(b) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}\right\}$.
(c) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}, x_{3}=x_{1}-x_{2}\right\}$.
(d) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{4}=x_{1}+x_{2}, x_{3}=x_{1}-x_{2}, x_{3}+x_{4}=2 x_{1}\right\}$.
(e) $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{1}=x_{2}=x_{3}=x_{4}\right\}$.

Victor Salazar
Victor Salazar
Numerade Educator
13:05

Problem 4

Determine the value of $\lambda \in \mathbb{R}$ for which each list of vectors is linear dependent.
(a) $((\lambda,-1,-1),(-1, \lambda,-1),(-1,-1, \lambda))$ as a subset of $\mathbb{R}^{3}$.
(b) $\sin 2(x), \cos (2 x), \lambda$ as a subset of $\mathcal{C}(\mathbb{R})$.

Vishvajeetkumar Bhaskar Batule
Vishvajeetkumar Bhaskar Batule
Numerade Educator
02:28

Problem 5

Consider the real vector space $V=\mathbb{R}^{4}$. For each of the following five statements, provide either a proof or a counterexample.
(a) $\operatorname{dim} V=4$.
(b) $\operatorname{span}((1,1,0,0),(0,1,1,0),(0,0,1,1))=V$.
(c) The list $((1,-1,0,0),(0,1,-1,0),(0,0,1,-1),(-1,0,0,1)$ )s linearly independent.
(d) Every list of four vectors $v_{1}, \ldots, v_{4} \in V$, such that $\operatorname{span}\left(v_{1}, \ldots, v_{4}\right)=V$, is linearly independent.
(e) Let $v_{1}$ and $v_{2}$ be two linearly independent vectors in $V$. Then, there exist vectors $u, w \in V$, such that $\left(v_{1}, v_{2}, u, w\right)$ is a basis for $V$.

Manisha Sarker
Manisha Sarker
Numerade Educator